Example 1: Torque Calculation for Index Table with Backlash

Table

The table is directly attached to the main shaft

Gears

Indexer

The cam motion-law is Mod-Sine, 180º index-period, and runs at 150RPM.  The cycle time is 0.4s with 0.2s motion and 0.2s dwell.

We can assume the cam-shaft runs at constant angular velocity.

There is an estimated overall Friction Torque of 2.0Nm in the output transmission referred to the indexer turret.

The maximum backlash permitted in the cam track, when manufactured is 0.04mm.

The backlash in the gears can wear to 0.1mm before adjustment or replacement.

Mass and Inertia of Components

The mass and inertia of each component are calculated as shown in the table above.

The final inertia (Combined Total) total in the above table takes into account the gear ratio.

The inertia of the slow-speed assembly is reduced by the square of the gear ratio when referred to the indexer turret shaft.

...an illustration of the benefits of a speed reduction!

Assessment of Dynamic System.

To assess the dynamic response, we estimate the natural vibration frequency of the system, and from that, the Period-Ratio.

The gears have a significant mass and inertia in an intermediate position in the transmission, which means that the system will not vibrate with a single, simple frequency, as required by the foregoing dynamic response theory!

However, the high inertia of the table and work stations imposes a dominant frequency which gives a good approximation to the theoretical model: this is nearly always the case in practice. But, be aware.

Rigidity

Torsional Rigidity of Shaft : ;

Bending Stiffness between bearings :

Rigidity: R = S × r2

Turret Shaft Torsional Rigidity

Second moment of area of shaft is:

Thus: Main Shaft Bending Stiffness is:

Note: Radius of Large Gear = 0.16m:

Its equivalent Torsional Rigidity is:

Indexer High Speed Shaft Torsional Rigidity

Turret Bending Stiffness is:

Note: Radius of Small Gear = 0.064m

Its equivalent Torsional Rigidity is:

The combined rigidity, referred to the indexer turret:

(Note that the bending stiffness is so high it could have been ignored).

Approximate Natural Frequency

Natural Period = 1/f = 0.0321 seconds (assuming a single degree of freedom).

Period-Ratio

The motion period is 0.2 seconds.

The Period-Ratio is:

n = Motion Period / Natural Period = T / (1/f) = f × T

n = 31.17 × 0.2 = 6.234

Nominal Peak Inertia Torque

The total inertia referred to the indexer is

Peak Angular Acceleration:

The output stroke of the indexer is:

Index-Period of the Indexer is:

The Coefficient of Acceleration of the Mod-Sine is:

Nominal Peak Acceleration is:

Peak Inertia Torque after Torque-Factor

Torque-Factor

We can use this equation to find the Torsion-Factor

;

The Parameters for the Mod-Sine Motion-Law are:

; ;

The Period-Ratio, n. is 6.34

Using the parameters, the Torsion-Factor is

We must increase the Nominal Peak Inertia Torque by the Torsion-Factor:

Add Friction Torque

We must add the Friction Torque for the mechanism, ignoring the backlash impact effect.

Include Backlash

We must consider the impact shock load after the transition of backlash...

Total Backlash, expressed as an angle (radians), at the Indexer Turret.

Gear Teeth, 0.1mm, @ 64mm radius:

Indexer Turret, 0.04mm @ 97.4mm radius:

Total Backlash

Normalized backlash (backlash against angular stroke)

Deceleration

The natural deceleration of the system due to the deceleration torque on the payload during 'Free-Flight' is:

Normalized Deceleration is:

This is quite low and can be taken as Zero!

Normalized Impact Velocity

With a Normalized Backlash of 0.00251 and Normalized Deceleration of 0.0, then:

Normalized Impact Velocity:

Real Impact Velocity:

Peak Shock Torque

The Peak Shock Torque on the Turret is:

At 40% of the Peak Inertia Torque without Backlash, this is a significant load, and should not be ignored in the design of the mechanism.

A safe way of taking it into account is simply to add it to the peak vibration torque (this assumes the two peak torques occur at exactly the same point in the motion, which is quite possible):

Peak Torque at Output Shaft.

Add the Peak Shock Torque to the Vibration Torque

Although it could be argued that this is too pessimistic, it does illustrate that to design the mechanism on the basis of the Nominal Dynamic Torque of 78.41Nm would be under-estimated!